What are some concrete steps to deal with the vanishing gradient problem?
There is not single answer to the vanishing gradient problem. However, there a few things that can help.

As mentioned in the comments, use of Rectified Linear Units (ReLU) as your activation function can help, since the it does not get saturated for large neuron inputs.

Next, careful choice of weight initialization can help avoid saturation, as well. See Andre Ng's Coursera video for details.

Finally, if you are concerned that the scale of your training input is causing issues with training, you can normalize the training examples by subtracting the mean and dividing by the standard deviation. There is a normality assumption here, but this can often help avoid problems where one feature is far out of scale with another. However, this usually causes issues with optimization bouncing between the walls of long, thin trough, which complicates convergence, but does not cause vanishing gradient.